For the most part yes. Just heard an email from one person on the radio who was quoted 5 months if they went public for brain surgery and went private and got it next day.
In my own case (and I accept this is not life-threatening) but we had to sort out speech therapy for one of our smallies a few years ago, if we waited for the state services we were looking at 18-24 months before we'd even get to see someone. Likewise on another occasion my better half needed a small (non-critical) operation, again an 18 month wait on public.. On both occasions we went private and both issues were being dealt with within a fortnight.
Having said that, I also had a smallie who recently needed dental surgery to remove a baby tooth which had no adult tooth growing underneath it. Again, not life threatening but it would never fall out and the gum would grow over it meaning if it was left the surgery would be more difficult. It was spotted as potentially an issue by the school dental check-ups, confirmed by our own dentist and operated on, in public within 6 weeks of the initial school check up. In fairness, I cannot fault the public system for that. We didn't go private since we got the appointment so quickly. We were asked at the hospital on the day did we have private insurance and said yes and filled out the forms to the hospital got paid for it but on this occasion having private made no difference.